We start by defining the functor \(Q^{\prime }\) .
Its construction is not specific to simplicial presheaves on \({{C}\mathrm {art}}\) , but it works for simplicial presheaves over any small category.
Thus, let \({C}\)  be a small category, let \({{K}}_\infty \)  denote the category of simplicial presheaves on \({C}\) , and let \({{Y}}\colon {C}\rightarrow {{K}}_\infty \)  denote the Yoneda embedding.
We denote the projective and the injective model structures on \({{K}}_\infty \)  by \({{K}}_\infty ^p\)  and by \({{K}}_\infty ^i\) , respectively.
The conventional two-sided simplicial bar construction provides a functor [1]}
\(B_\bullet \big ( (-), {C}, {{Y}}\big ) \colon {{K}}_\infty \longrightarrow ({{K}}_\infty )^{{\mathbbe {\Delta }}^\mathrm {op}}\,,\qquad F \longmapsto B_\bullet (F,{C},{{Y}})\,.\) 
